A reasonable way to model a donut is by revolving a region R, Whose boundary is a circle, about the yaxis. I I y I I A rendition of the donut The region R in the myplane Some actual measurements we can take from the donut that we can use for the modeling include the following. o The distance a from the center of the hole in the donut to the inner ring. a The distance b from the center of the hole in the donut to the outer ring. A. [3 pts] Find an equation for the circle that forms the boundary of R. Leave a and b in your nal answer. B. [3 pts] Often times, the measurements that we can reasonably find lead to notationally cumbersome equations in our modeling or we can note that certain quantities appear frequently in the model. In these instances, it is often helpful to introduce auxiliary variables to make the notation look cleaner . Set h = a +b b - a and r = 2 2 -, and deduce that the equation of the circle in Part A becomes (x - h)+y? = 2.C. [10 pts] Use the washer method to Show that the volume of the donut is given by the integral below. V=/:1r[(h+\\/1Tyz)2- (hW)2]dy In order to receive full credit, you must draw an appropriate horizontal or vertical slice on the image below that, when revolved, forms a. washer and label the inner and outer radii. y D. [5 pts] Simplify the volume integral above. Then, explain how symmetry can be used to conclude that V =f Swh'irz y? dy. u E. [12 pts] Use an appropriate trigonometric substitution to evaluate the integral V = f Sum/1'2 y2 (11;. 0 Leave your nal answer in terms of r and h, and simplify it completely. F. [7 pts] A standard graphing calculator cannot evaluate integrals with unspecied parameters like r and h. However, such a calculator can be used to check your work. One method is as follows. Suppose that we measure that a: = 3 and l} = 6, where a and b are the measurements dened on the previous page. Do the following. i. Calculate 1' and h when a = 3 and b = 6 and write them in the space below. T: h: ii. Technology can be used to evaluate f 87rhx/ 1'2 '92 dz; for the values for 'r and it you found. Plug in these values, then ll in the sentence btelow. The integral that I am evaluating is . To evaluate this integral, I used and found that to 3 decimal places, the volume of the donut is iii. Substitute the values for 7' and h into your answer for Part E and write down this expression. Then, use technology to evaluate it to 3 decimal places. Note that if you simply write down the result from ii. and it is not consistent with your response to Part E, you will receive 0 points for this part. iv. Are your answers to Parts ii. and iii. consistent? Explain they they should or should not be